Lerche–Newberger sum rule explained

The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982,^{[1] [2] [3]} finds the sum of certain infinite series involving Bessel functions J_α of the first kind. It states that if μ is any non-integer complex number,

\scriptstyle\gamma \in ( 0 , 1 ] , and Re(α + β) > -1, then\sum ^{inf in (( -1 ) ^{n} J _{\alpha} (z)J _{\beta} (z))/(n + \mu)} _{n =-inf in} = (\pi)/(\sin \mu \pi) J _{\alpha} (z)J _{\beta} (z).

Newberger's formula generalizes a formula of this type proven by Lerche in 1966; Newberger discovered it independently. Lerche's formula has γ =1; both extend a standard rule for the summation of Bessel functions, and are useful in plasma physics.^{[4] [5] [6] [7]}

Notes and References

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